forecasting using discrete event simulation for the nz prison population dr jason (qingsheng) wang...
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Forecasting using Discrete Event Simulation for the NZ Prison Population
Dr Jason (Qingsheng) Wang
Mr Ross Edney
Ministry of Justice
2
Prison Population Projections
• The prison population in NZ has grown steadily over the last decade
• The prisoner population is the end result of a chain of complex causal factors
• Serious crime, apprehensions, investigations, prosecutions and court cases which can involve variable periods of remand are precursors to imprisonment
• The opportunity cost of excess prison beds is high
• The costs of bed shortages is also high due in part to risks to staff and prisoner safety
• Several years of planning is required to build a prison so accurate forecasts are important
3
Discrete Event Simulation (DES)
• DES treats each prisoner entering the prison system separately and assigns the time that will be spent in prison so there is an entry date and exit date
• The technique “releases” prisoners from the virtual prison system when they have served their sentence
• DES can support multiple inflows • A forecast track for the prisoner population can be
derived by using the inflow and stay-time data
4
Remand Prison Population DES Model and Its Direct Drivers
• Remand prisoner inflows/arrivals
• Time spend on remand
month
start
t y
Set A(5)
daily rate
F
Rmonth
Remand
CT SF
0
numbersinRemand Inflow
Time on Remand
F
R
Remand Prison
A
Get
Remand
F
Rmonth
3
RA
Exit#
171318DI
LUF PW
5
Sentenced Prison Population DES Model and Its Direct Drivers
• Sentenced prisoner inflows/arrivals
• Given sentence for each arrived prisoners
• Proportion of sentence served
month
start
t y
A: <=3M
Set A(5)
Given sent length
F
R
month
A: <=3M
CT SF
0
Proportion served
F
R
A
Get
Daily rate
month
A: <=3M
F
RF
R
Sentenced Prison
Sentenced Inflow A
ChangeA
month
A: <=3M
RA
Exit#
119209DI
LUF PW
3
6
Determining Forecast Assumptions
• Historic data is employed to derive key input assumptions
• Time series and expert opinion is used to form a view of the likely direction of key driver variables
• Experts provide independent input on the impact of new policy that may affect, for example, custodial sentences and their length during the forecast horizon
7
2006 Forecast, Sentenced Inflow
Inflows/Arrivals by Sentence GroupSentenced, Male
0
100
200
300
400
500
600
700
Jun-
98
Dec
-98
Jun-
99
Dec
-99
Jun-
00
Dec
-00
Jun-
01
Dec
-01
Jun-
02
Dec
-02
Jun-
03
Dec
-03
Jun-
04
Dec
-04
Jun-
05
Dec
-05
Jun-
06
Dec
-06
Jun-
07
Dec
-07
Jun-
08
Dec
-08
Jun-
09
Dec
-09
Jun-
10
Dec
-10
Jun-
11
Dec
-11
Jun-
12
Dec
-12
Jun-
13
Dec
-13
Jun-
14
Arrival, <=1Y Arrival, >1Y to 2Y Arrival, >2Y Arrival, SV
Inflow/Arrivals Time-Series Projections
8
2006 Forecast, Proportion Served
Projected Proportion Served (excl Remand) for > 2Y Non-SV, Male
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Jun-
98
Oct
-98
Feb
-99
Jun-
99
Oct
-99
Feb
-00
Jun-
00
Oct
-00
Feb
-01
Jun-
01
Oct
-01
Feb
-02
Jun-
02
Oct
-02
Feb
-03
Jun-
03
Oct
-03
Feb
-04
Jun-
04
Oct
-04
Feb
-05
Jun-
05
Oct
-05
Feb
-06
Jun-
06
Oct
-06
Feb
-07
Jun-
07
Oct
-07
Feb
-08
Jun-
08
Oct
-08
Feb
-09
Jun-
09
Oct
-09
Feb
-10
Jun-
10
>2Y to 3Y >3Y to 5Y >5Y Proj E: >2Y to 3Y Proj F: >3Y to 5Y Proj G: >5Y
9
2006 Forecast, Male Sentenced
Simulation Projections with Assumptions of Baseline Scenario Sentenced Male
0
500
1000
1500
2000
2500
Jun-
98
Dec
-98
Jun-
99
Dec
-99
Jun-
00
Dec
-00
Jun-
01
Dec
-01
Jun-
02
Dec
-02
Jun-
03
Dec
-03
Jun-
04
Dec
-04
Jun-
05
Dec
-05
Jun-
06
Dec
-06
Jun-
07
Dec
-07
Jun-
08
Dec
-08
Jun-
09
Dec
-09
Jun-
10
Dec
-10
Jun-
11
Dec
-11
Jun-
12
Dec
-12
Jun-
13
Dec
-13
Jun-
14
Muster, <=1Y Muster, >1Y to 2Y Muster, >2Y Muster, SV Muster, Life&PD
Sim Proj Baseline, <=1Y Sim Proj Baseline, >1Y to 2Y Sim Proj Beseline, >2Y Sim Proj Baseline, SV Time-series Proj, Life&PD
10
2006 Forecast, Male Remand
Simulation Projection with Assumptions of Baseline Scenario Remand Male
0
500
1,000
1,500
2,000
2,500
3,000
Jun-
98
Dec
-98
Jun-
99
Dec
-99
Jun-
00
Dec
-00
Jun-
01
Dec
-01
Jun-
02
Dec
-02
Jun-
03
Dec
-03
Jun-
04
Dec
-04
Jun-
05
Dec
-05
Jun-
06
Dec
-06
Jun-
07
Dec
-07
Jun-
08
Dec
-08
Jun-
09
Dec
-09
Jun-
10
Dec
-10
Jun-
11
Dec
-11
Jun-
12
Dec
-12
Jun-
13
Dec
-13
Jun-
14
Remand Muster, Male Simulation, Baseline Proj, Male
11
2006 Forecast Performance,Including EI after New Sentences
Total Prison Population
6000
6500
7000
7500
8000
8500
9000
9500
1-J
an-0
71
5-J
an
-07
29-J
an
-07
12-F
eb
-07
26-F
eb
-07
12-M
ar-
07
26-M
ar-
07
9-A
pr-
07
23-A
pr-
07
7-M
ay-
07
21-M
ay-
07
4-J
un-0
71
8-J
un
-07
2-J
ul-0
71
6-J
ul-
07
30-J
ul-
07
13-A
ug-0
72
7-A
ug-0
71
0-S
ep-0
72
4-S
ep-0
78
-Oct
-07
22-O
ct-0
75
-No
v-07
19-N
ov-
07
3-D
ec-
07
17-D
ec-
07
31-D
ec-
07
14-J
an
-08
28-J
an
-08
11-F
eb
-08
25-F
eb
-08
10-M
ar-
08
24-M
ar-
08
7-A
pr-
08
21-A
pr-
08
5-M
ay-
08
19-M
ay-
08
2-J
un-0
81
6-J
un
-08
30-J
un
-08
14-J
ul-
08
28-J
ul-
08
11-A
ug-0
82
5-A
ug-0
88
-Se
p-0
82
2-S
ep-0
86
-Oct
-08
20-O
ct-0
83
-No
v-08
17-N
ov-
08
1-D
ec-
08
15-D
ec-
08
29-D
ec-
08
Date
Pri
so
ner
da
ily c
ou
nt
Maximum Justice beds available to Corrections (Police Cells)
Maximum Corrections operating capacity
Total actual Prison population
Average daily operating capacity
2006 Justice Sector Forecast
Monday peak 7,848
Spring Hill opens - phased occupancy
12
DES Monte-Carlo Model, Confidence Intervals, and Capacity Planning Buffer
• Instead of using fixed mean values, the distributional DES model will feed the direct drivers with distributional values.
• Monte Carlo simulation will generate the population/muster probability distribution so to identify peak demand and risks over the planning horizon
• Policy simulation: this approach can be applied in policy changes with distributional impact
13
Sentenced Distributional DES Model
F
R
CT SF
year
Time on remand distributions
Set A(5)
Input table:
daily rate
Remand Male
Remand Inflow distributions
Remand Prison
F
R
month
1
F
Rmonth
3
RA
A
Get
DI
LUF PW
Exit#
190035
F
R
change
run#
month
Writes results
14
Remand Monthly Inflow and Distribution
• Using normal distribution of Holt-Winters model residuals and prediction errors
-500
0
500
1,000
1,500
2,000
2,500
1-J
un
-98
1-D
ec-9
8
1-J
un
-99
1-D
ec-9
9
1-J
un
-00
1-D
ec-0
0
1-J
un
-01
1-D
ec-0
1
1-J
un
-02
1-D
ec-0
2
1-J
un
-03
1-D
ec-0
3
1-J
un
-04
1-D
ec-0
4
1-J
un
-05
1-D
ec-0
5
1-J
un
-06
1-D
ec-0
6
1-J
un
-07
1-D
ec-0
7
1-J
un
-08
1-D
ec-0
8
1-J
un
-09
1-D
ec-0
9
1-J
un
-10
1-D
ec-1
0
1-J
un
-11
1-D
ec-1
1
1-J
un
-12
1-D
ec-1
2
1-J
un
-13
1-D
ec-1
3
1-J
un
-14
1-D
ec-1
4
1-J
un
-15
RESIDUAL
ACTUAL
FORECAST
STD
L95
U95
Drop Page Fields Here
Sum of MRem_Inflow
Date
_TYPE_
15
Time on Remand Distributions• Annual empirical distributions
Projection o f T im e on Rem and Distributions, M ale
0%
5%
10%
15%
20%
25%
30%
35%
01:Up to 07days
02:Up to 14days
03:Up to 21days
04:Up to 28days
05:Up to 35days
06:Up to 49days
07:Up to 84days
08:Up to 189days
09:Up to 364days
10:Morethan 364
days
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
16
Remand Muster Monte-Carlo, Male
Male Remand Distributional Model's Monte-Carlo v3, Inflow-Normal & Time-on-Empirical
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Oct-95 Jul-98 Apr-01 Jan-04 Oct-06 Jul-09 Apr-12 Dec-14 Sep-17
1 2 34 5 67 8 910 11 1213 14 1516 17 1819 20 2122 23 2425 26 2728 29 3031 32 3334 35 3637 38 3940 41 4243 44 4546 47 4849 50 5152 53 5455 56 5758 59 6061 62 6364 65 6667 68 6970 71 7273 74 7576 77 7879 80 8182 83 8485 86 8788 89 9091 92 9394 95 9697 98 99100 101 102103 104 105106 107 108109 110 111112 113 114115 116 117118 119 120121 122 123124 125 126127 128 129130 131 132133 134 135136 137 138139 140 141142 143 144145 146 147148 149 150
17
Muster Distribution Normality TestCross-Section Distributions, Remand
500 Runs of Male Remand Distributional Model v3 based on Inflow-Normal & Time-on-Empirical Distributions
Std Kurtosis Skewness Jarque-Bera testχ²(JB,2) Remand muster cross section distribution at July-199937.03036 0.060376 0.040828 0.21485 0.898144
Bin FrequencyCumulative %Normal Cumulative %Rem Dis Normal Dis594 2 0.4% 0% 0% 0%619 9 2.2% 2% 2% 2%644 33 8.8% 9% 7% 7%669 86 26.0% 25% 17% 16%694 117 49.4% 50% 23% 25%719 137 76.8% 75% 27% 25%744 68 90.4% 91% 14% 16%769 39 98.2% 98% 8% 7%794 7 99.6% 100% 1% 2%819 1 99.8% 100% 0% 0%
More 1 100.0%
Std Kurtosis Skewness Jarque-Bera testχ²(JB,2) Remand muster cross section distribution at July-200039.03053 -0.045201 -0.169691 2.442145 0.294914
Bin FrequencyCumulative %Normal Cumulative %Rem Dis Normal Dis704 2 0.4% 0% 0% 0%730 14 3.2% 2% 3% 2%756 37 10.6% 9% 7% 7%782 69 24.4% 25% 14% 16%808 123 49.0% 50% 25% 25%834 130 75.0% 75% 26% 25%860 85 92.0% 91% 17% 16%886 31 98.2% 98% 6% 7%912 7 99.6% 100% 1% 2%938 2 100.0% 100% 0% 0%
More 0 100.0%
Std Kurtosis Skewness Jarque-Bera testχ²(JB,2) Remand muster cross section distribution at July-200138.70658 -0.123777 0.08981 0.991343 0.609162
Bin FrequencyCumulative %Normal Cumulative %Rem Dis Normal Dis733 2 0.4% 0% 0% 0%758 8 2.0% 3% 2% 2%783 38 9.6% 10% 8% 7%808 91 27.8% 26% 18% 16%833 124 52.6% 50% 25% 24%858 109 74.4% 74% 22% 24%883 83 91.0% 90% 17% 16%908 31 97.2% 97% 6% 7%933 11 99.4% 100% 2% 2%958 2 99.8% 100% 0% 0%
More 1 100.0%
0%
5%
10%
15%
20%
25%
30%
1 2 3 4 5 6 7 8 9 10
Rem Dis Normal Dis
0%
5%
10%
15%
20%
25%
30%
1 2 3 4 5 6 7 8 9 10
Rem Dis Normal Dis
0%
5%
10%
15%
20%
25%
30%
1 2 3 4 5 6 7 8 9 10
Rem Dis Normal Dis
18
Standardised Monte-Carlo, Remand
Normalised Monte-Carlo, Remand Male v3 (Inflow-Normal & Time-On-Empirical)
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
Oct-95 Jul-98 Apr-01 Jan-04 Oct-06 Jul-09 Apr-12 Dec-14 Sep-17
1 23 45 67 89 1011 1213 1415 1617 1819 2021 2223 2425 2627 2829 3031 3233 3435 3637 3839 4041 4243 4445 4647 4849 5051 5253 5455 5657 5859 6061 6263 6465 6667 6869 7071 7273 7475 7677 7879 8081 8283 8485 8687 8889 9091 9293 9495 9697 9899 100
19
Standardised Monte-Carlo’s Distributions, Remand
Normalised Monte-Carlo, Remand Male v3 (Inflow-Normal & Time-On-Empirical)
From Jun-98 to Aug-07
Std Kurtosis Skewness Jarque-Bera testχ²(JB,2)0.99901 -0.004549 0.0323931 9.753995 0.0076199
BinObservatio
ns
Rem Cumulative
%
Normal Cumulative
%Rem Dis Normal Dis
-4 1 0.0% 0.0% 0.0% 0.0%-3 63 0.1% 0.1% 0.1% 0.1%-2 1,142 2.2% 2.3% 2.1% 2.1%-1 7,605 15.9% 15.8% 13.7% 13.6%0 19,108 50.3% 50.0% 34.4% 34.2%1 18,696 84.0% 84.2% 33.7% 34.2%2 7,602 97.7% 97.7% 13.7% 13.6%3 1,203 99.9% 99.9% 2.2% 2.1%4 79 100.0% 100.0% 0.1% 0.1%5 1 100.0% 100.0% 0.0% 0.0%
More 0 100%55,500
From Sep-07 to Aug-15
Std Kurtosis Skewness Jarque-Bera testχ²(JB,2)0.99901 0.0307374 -0.0024895 1.98 0.3716581
BinObservatio
ns
Rem Cumulative
%
Normal Cumulative
%Rem Dis Normal Dis
-4 3 0.0% 0.0% 0.0% 0.0%-3 72 0.2% 0.1% 0.2% 0.1%-2 1,013 2.3% 2.3% 2.1% 2.1%-1 6,523 15.9% 15.8% 13.6% 13.6%0 16,345 49.9% 50.0% 34.1% 34.2%1 16,422 84.1% 84.2% 34.2% 34.2%2 6,524 97.7% 97.7% 13.6% 13.6%3 1,034 99.9% 99.9% 2.2% 2.1%4 63 100.0% 100.0% 0.1% 0.1%5 1 100.0% 100.0% 0.0% 0.0%
More 0 100.0%48,000
-5.0%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
1 2 3 4 5 6 7 8 9 10
Rem Dis Normal Dis
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
1 2 3 4 5 6 7 8 9 10
Rem Dis Normal Dis
20
Standardised Monte-Carlo Distribution, Sentenced
Normalised Monte-Carlo, Sentenced Male v2 (Inflow Normal, Given Sentence &PropServ Empirical)
From Jun-98 to Aug-07
Std Kurtosis Skewness Jarque-Bera testχ²(JB,2)0.999008 -0.079553 0.02493332 20.385624 3.74385E-05
BinObservatio
ns
Sent Cumulative
%
Normal Cumulative
%Sent Dis Normal Dis
-4 3 0.01% 0.0% 0.01% 0.00%-3 41 0.08% 0.1% 0.07% 0.13%-2 1161 2.17% 2.3% 2.09% 2.13%-1 7646 15.95% 15.8% 13.78% 13.58%0 19017 50.21% 50.0% 34.26% 34.16%1 18701 83.91% 84.2% 33.70% 34.16%2 7669 97.73% 97.7% 13.82% 13.58%3 1190 99.87% 99.9% 2.14% 2.13%4 69 99.99% 100.0% 0.12% 0.13%5 3 100.00% 100.0% 0.01% 0.00%
More 0 100.00%55,500
1.000972714
From Sep-07 to Aug-15
Std Kurtosis Skewness Jarque-Bera testχ²(JB,2)0.99901 -0.013876 -0.0488389 19.47 5.93E-05
BinObservatio
ns
Sent Cumulative
%
Normal Cumulative
%Sent Dis Normal Dis
-4 8 0.02% 0.0% 0.0% 0.0%-3 67 0.16% 0.1% 0.1% 0.1%-2 1057 2.36% 2.3% 2.2% 2.1%-1 6592 16.09% 15.8% 13.7% 13.6%0 15978 49.38% 50.0% 33.3% 34.2%1 16659 84.09% 84.2% 34.7% 34.2%2 6523 97.68% 97.7% 13.6% 13.6%3 1086 99.94% 99.9% 2.3% 2.1%4 30 100.00% 100.0% 0.1% 0.1%5 0 100.00% 100.0% 0.0% 0.0%
More 0 100.00%48,000
1.2815516
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
30.00%
35.00%
40.00%
-4 -3 -2 -1 0 1 2 3 4 5
Sent Dis Normal Dis
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
1 2 3 4 5 6 7 8 9 10
Sent Dis Normal Dis
Normal Dis
0%
5%
10%
15%
20%
25%
30%
35%
40%
-4 -3 -2 -1 0 1 2 3 4 5
Normal Dis
21
Forecast Confidence Intervals with Normal Distribution and Empirical Distribution
Mean Forecast with 90% Confidence Intervals of Normal & Empirical Distributions, Male Total
6,500
7,000
7,500
8,000
8,500
9,000
9,500
10,000
10,500
11,000
11,500
Sep
-07
Dec
-07
Mar
-08
Jun-
08
Sep
-08
Dec
-08
Mar
-09
Jun-
09
Sep
-09
Dec
-09
Mar
-10
Jun-
10
Sep
-10
Dec
-10
Mar
-11
Jun-
11
Sep
-11
Dec
-11
Mar
-12
Jun-
12
Sep
-12
Dec
-12
Mar
-13
Jun-
13
Sep
-13
Dec
-13
Mar
-14
Jun-
14
Sep
-14
Dec
-14
Mar
-15
Jun-
15
Male Musters Mean Forecast Upper 90%, Normal Dis Lower 90%, Normal Dis Upper 90%, Empirical Dis Lower 90%, Empirical Dis
22
Volatility Buffer Calculation Assuming Normal Distributions
Male Prison Bed's Buffer Calculation, 1 STD for Optimal Pricing of BS Model
0
100
200
300
400
500
600
700
800
Sep
-07
Dec
-07
Mar
-08
Jun-
08
Sep
-08
Dec
-08
Mar
-09
Jun-
09
Sep
-09
Dec
-09
Mar
-10
Jun-
10
Sep
-10
Dec
-10
Mar
-11
Jun-
11
Sep
-11
Dec
-11
Mar
-12
Jun-
12
Sep
-12
Dec
-12
Mar
-13
Jun-
13
Sep
-13
Dec
-13
Mar
-14
Jun-
14
Sep
-14
Dec
-14
Mar
-15
Jun-
15
0%
1%
2%
3%
4%
5%
6%
7%
8%
Rem Buffer Calculation (1 Std) Sent Buffer Calculation (1 Std) % Buffer Calculation
23
End
•Thank you
24
Jarque-Bera Test, Remand
Jarque-Bera Normality Test, Male Remand v3 (Inflow-Normal & Time-On-Empirical)
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
Jun-
98
Dec-9
8
Jun-
99
Dec-9
9
Jun-
00
Dec-0
0
Jun-
01
Dec-0
1
Jun-
02
Dec-0
2
Jun-
03
Dec-0
3
Jun-
04
Dec-0
4
Jun-
05
Dec-0
5
Jun-
06
Dec-0
6
Jun-
07
Dec-0
7
Jun-
08
Dec-0
8
Jun-
09
Dec-0
9
Jun-
10
Dec-1
0
Jun-
11
Dec-1
1
Jun-
12
Dec-1
2
Jun-
13
Dec-1
3
Jun-
14
Dec-1
4
Jun-
15
?²(JB,2)
25
Sentenced Distributional DES ModelInflows and Distributions
-200
-100
0
100
200
300
400
500
600
700
800
900
1-J
un-9
8
1-D
ec-
98
1-J
un-9
9
1-D
ec-
99
1-J
un-0
0
1-D
ec-0
0
1-Ju
n-0
1
1-D
ec-0
1
1-Ju
n-0
2
1-D
ec-0
2
1-Ju
n-03
1-D
ec-0
3
1-Ju
n-04
1-D
ec-0
4
1-Ju
n-05
1-D
ec
-05
1-J
un-
06
1-D
ec
-06
1-J
un-
07
1-D
ec-
07
1-J
un-0
8
1-D
ec-
08
1-J
un-0
9
1-D
ec-
09
1-J
un-1
0
1-D
ec-
10
1-J
un-1
1
1-D
ec-
11
1-J
un-1
2
1-D
ec-1
2
1-Ju
n-1
3
1-D
ec-1
3
1-Ju
n-1
4
1-D
ec-1
4
1-Ju
n-15
1-D
ec-1
5
RESIDUAL
ACTUAL
FORECAST
STD
L95
U95
Drop Page Fields Here
Sum of Le1Y
sentgp2
_TYPE_
<=1Y Forecast with Step Changes and Multiplicative Seasonality
0
100
200
300
400
500
600
Jun-
98
Dec
-98
Jun-
99
Dec
-99
Jun-
00
Dec
-00
Jun-
01
Dec
-01
Jun-
02
Dec
-02
Jun-
03
Dec
-03
Jun-
04
Dec
-04
Jun-
05
Dec
-05
Jun-
06
Dec
-06
Jun-
07
Dec
-07
Jun-
08
Dec
-08
Jun-
09
Dec
-09
Jun-
10
Dec
-10
Jun-
11
Dec
-11
Jun-
12
Dec
-12
Jun-
13
Dec
-13
Jun-
14
Dec
-14
Jun-
15
Dec
-15
Actual New forecast w ith step changes
-100
-50
0
50
100
150
200
250
300
1-Ju
n-98
1-D
ec-9
8
1-Ju
n-99
1-D
ec-9
9
1-Ju
n-00
1-D
ec-0
0
1-Ju
n-01
1-D
ec-0
1
1-Ju
n-02
1-D
ec-0
2
1-Ju
n-03
1-D
ec-0
3
1-Ju
n-04
1-D
ec-0
4
1-Ju
n-05
1-D
ec-0
5
1-Ju
n-06
1-D
ec-0
6
1-Ju
n-07
1-D
ec-0
7
1-Ju
n-08
1-D
ec-0
8
1-Ju
n-09
1-D
ec-0
9
1-Ju
n-10
1-D
ec-1
0
1-Ju
n-11
1-D
ec-1
1
1-Ju
n-12
1-D
ec-1
2
1-Ju
n-13
1-D
ec-1
3
1-Ju
n-14
1-D
ec-1
4
1-Ju
n-15
1-D
ec-1
5
RESIDUAL
ACTUAL
FORECAST
STD
L95
U95
Drop Page Fields Here
Sum of Y1to2Y
sentgp2
_TYPE_
>1Y to 2Y Forecast with Step Changes and Multiplicative Seasonality
0
20
40
60
80
100
120
140
160
180
200
Jun-
98
Dec
-98
Jun-
99
Dec
-99
Jun-
00
Dec
-00
Jun-
01
Dec
-01
Jun-
02
Dec
-02
Jun-
03
Dec
-03
Jun-
04
Dec
-04
Jun-
05
Dec
-05
Jun-
06
Dec
-06
Jun-
07
Dec
-07
Jun-
08
Dec
-08
Jun-
09
Dec
-09
Jun-
10
Dec
-10
Jun-
11
Dec
-11
Jun-
12
Dec
-12
Jun-
13
Dec
-13
Jun-
14
Dec
-14
Jun-
15
Dec
-15
Actual New forecast w ith step changes
26
Sentenced Distributional DES ModelGiven Sentence Distributions
The Emprical Distributions of Given Sentences for <=1Y by Jun-Years
0%
5%
10%
15%
20%
25%
01: 1M, 31Days
02: 2M, 62Days
03: 3M, 92Days
04: 4M, 123Days
05: 5M, 153Days
06: 6M, 184Days
07: 7M, 214Days
08: 8M, 245Days
09: 9M, 276Days
10: 10M,306 Days
11: 11M,337 Days
12: 12M,366 Days
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Grand Total
The Emprical Distributions of Given Sentences for >1Y to 2Y by Jun-Years
0%
5%
10%
15%
20%
25%
30%
35%
13: 1Y1M,397 Days
14: 1Y2M,427 Days
15: 1Y3M,458 Days
16: 1Y4M,489 Days
17: 1Y5M,519 Days
18: 1Y6M,550 Days
19: 1Y7M,580 Days
20: 1Y8M,611 Days
21: 1Y9M,641 Days
22: 1Y10M,671 Days
23: 1Y11M,701 Days
24: 2Y, 731Days
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Grand Total
The Emprical Distributions of Given Sentences for >2Y Non-SV by Jun-Years
0%
10%
20%
30%
40%
50%
60%
70%
25: 3Y, 1096Days
26: 4Y, 1461Days
27: 5Y, 1827Days
28: 6Y, 2192Days
29: 7Y, 2557Days
30: 8Y, 2922Days
31: 9Y, 3288Days
32: 10Y, 3653Days
33: >10Y
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Grand Total
The Emprical Distributions of Given Sentences for >2Y SV by Jun-Years
0%
5%
10%
15%
20%
25%
30%
35%
25: 3Y, 1096Days
26: 4Y, 1461Days
27: 5Y, 1827Days
28: 6Y, 2192Days
29: 7Y, 2557Days
30: 8Y, 2922Days
31: 9Y, 3288Days
32: 10Y, 3653Days
33: >10Y
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Grand Total
27
Sentenced Distributional DES ModelProportion Served Distributions
Proportion Sentence Served excluding Remand, Male
0%
10%
20%
30%
40%
50%
60%
70%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
01: <=1Y 02: >1Y to 2Y 03: >2Y 04: SV
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Gender2 Male+
Sum of Percent
sentgp2 PropServ_asSent
End_JuneY
28
Proportion Served Distributions
>2Y. Non-SV
-5%
0%
5%
10%
15%
20%
25%
30%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
SV, >2Y
-10%
0%
10%
20%
30%
40%
50%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
29
Sentenced Muster Monte-Carlo, Male
Male Sentenced Distributional Model's Monte-Carlo
0
1000
2000
3000
4000
5000
6000
7000
8000
Oct-95 Jul-98 Apr-01 Jan-04 Oct-06 Jul-09 Apr-12 Dec-14 Sep-17
1 2 34 5 67 8 910 11 1213 14 1516 17 1819 20 2122 23 2425 26 2728 29 3031 32 3334 35 3637 38 3940 41 4243 44 4546 47 4849 50 5152 53 5455 56 5758 59 6061 62 6364 65 6667 68 6970 71 7273 74 7576 77 7879 80 8182 83 8485 86 8788 89 9091 92 9394 95 9697 98 99100 101 102103 104 105106 107 108109 110 111112 113 114115 116 117118 119 120121 122 123124 125 126127 128 129130 131 132133 134 135136 137 138139 140 141142 143 144145 146 147148 149 150
30
Jarque-Bera Test, Sentenced Male
Jarque-Bera Test, Male Sentenced v3 (Inflow-Normal, Given Sentence & PropServ Empirical)
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
Jun-
98
Dec-9
8
Jun-
99
Dec-9
9
Jun-
00
Dec-0
0
Jun-
01
Dec-0
1
Jun-
02
Dec-0
2
Jun-
03
Dec-0
3
Jun-
04
Dec-0
4
Jun-
05
Dec-0
5
Jun-
06
Dec-0
6
Jun-
07
Dec-0
7
Jun-
08
Dec-0
8
Jun-
09
Dec-0
9
Jun-
10
Dec-1
0
Jun-
11
Dec-1
1
Jun-
12
Dec-1
2
Jun-
13
Dec-1
3
Jun-
14
Dec-1
4
Jun-
15
?²(JB,2)